Applications of variational models in geometric problems

Variational level set and PDE based methods and their applications in digital image processing have been well developed and studied for the past twenty years. These methods were soon applied to geometric problems. In this thesis, various geometric problems including surface denoising, surface reconstruction, image segmentation with priors are extensively studied. Especially with recently developed fast optimization method so called split Bregman[97], many of these problems got significant improvement over existing methods in speed.The first topic is on surface restoration using nonlocal means [1], where we extend nonlocal smoothing techniques for image regularization in [12] to surface regularization, with surfaces represented by level set functions. Numerical results show that our extension of nonlocal smoothing to surface regularization is very effective in removing spurious oscillations while preserving and even restoring sharp features. Furthermore, topology corrections are also made by our algorithms for some of the surfaces.The second topic is on point clouds surface reconstruction by using two different methods. The first one is to use multigrid narrow band methods via level sets. The computation constrained in narrow band greatly speeds up the algorithm. The second one is to use fast split Bregman methods on solving a convexified model and Bregman iteration for final reconstruction. Numerical results are presented to show the accuracy, efficiency and robustness of our method in reconstructing a wide variety of shapes.The last, but not the least, topic is on hyperspectral image segmentation with priors. The priors can be signal prior or shape prior. The results are greatly improved with the help of priors from the demonstration for both synthetic images and hyperspectral images. The use of split Bregman algorithm also makes our method very efficient comparing other existing segmentation methods with priors.